Boundary Regularity and Uniqueness for Minimal Sets

• Applicant: Dr. Simone Steinbrüchel, Ph.D.
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 522292454

This project consists of two parts. The main part is to analyze minimizing sets at their boundary. Minimizing sets originate in the study of soap films by observing that when pushing a wire into soapsuds, the surface spanned by the soap uses the least possible area while it has to span the boundary given by the wire. There are different mathematical models to describe the minimization problem. We focus on a set theoretic one proposed by Reifenberg and Taylor who studied the behaviour of this minimal set away from its boundary. Our project however focuses on the minimal set near its boundary. It is known that singularities, such as splitting into three different planes, can occur and we want to investigate how they look like and estimate how many there are. To this end, we first study their first order approximation which are cones. We aim to classify all possible minimal cones which then gives information about the original minimizing set. In a second part, we are interested in the uniqueness of the minimizers, i.e. whether it is true that if one puts the same wire several times into soapsuds, the spanned surface is always the same. The general answer is known to be negative, however those boundaries that lead to several minimizers carry a lot of symmetry. We thus want to quantify how many boundaries have a unique minimizing set. We conjecture that this is a very small quantity, in fact we think that the set of such boundaries is what in set theory is called meager.

• DFG Programme: Research Grants